The answer
(a)(i) \(T_n = 59 - 4n\)
(ii) \(-41\)
(b)(i) \(n + 9\)
(ii) shown
(iii) shown
O-Level E-Math 2015 Paper 2 Question 8 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 8 of the O-Level E-Math 2015 Paper 2. It tests nth term of an arithmetic sequence, in the Sequences / number patterns area. It is worth 9 marks: (a) 2 + 1, (b) 1 + 2 + 3. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a)(i) First term 55, common difference \(-4\): \(T_n = 55 + (n-1)(-4) = 59 - 4n\). (Check: \(n=1 \to 55\), \(n=4 \to 43\).)
(ii) \(T_{25} = 59 - 100 = -41\).
(b)(i) The grid has 8 columns, so the number below any number is 8 more. The square is \(\begin{smallmatrix} n & n+1 \\ n+8 & n+9 \end{smallmatrix}\), so the bottom-right number is \(n + 9\).
(ii) Products of opposite corners: \((n+1)(n+8) = n^2 + 9n + 8\) and \(n(n+9) = n^2 + 9n\). Difference \(= (n^2 + 9n + 8) - (n^2 + 9n) = 8\), which is independent of \(n\). (shown)
(iii) Sum \(= n + (n+1) + (n+8) + (n+9) = 4n + 18\). If this equalled 260, then \(4n = 242 \Rightarrow n = 60.5\), which is not a whole number. Since \(n\) must be a whole number on the grid, the sum cannot be 260. (shown)
Answer: (a)(i) \(T_n = 59 - 4n\)
(ii) \(-41\)
(b)(i) \(n + 9\)
(ii) shown
(iii) shown
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a nth term of an arithmetic sequence question from Sequences / number patterns, worth 9 marks: (a) 2 + 1, (b) 1 + 2 + 3.
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